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DOI: 10.1007/s11118-014-9458-x

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Citation for published version (APA): Riedle, M. (2015). Ornstein-Uhlenbeck Processes Driven by Cylindrical Lévy Processes. POTENTIAL ANALYSIS, 42(4), 809-838. 10.1007/s11118-014-9458-x

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Download date: 18. Feb. 2017 Ornstein-Uhlenbeck processes driven by cylindrical L´evyprocesses

Markus Riedle∗ Department of Mathematics King’s College London WC2R 2LS United Kingdom December 19, 2014

Abstract In this article we introduce a theory of integration for deterministic, operator-valued integrands with respect to cylindrical Lévyprocesses in separable Banach spaces. Here, a cylindrical Lévyprocess is understood in the classical framework of cylindrical random variables and cylindrical measures, and thus, it can be considered as a natural generalisation of cylindrical Wiener processes or white noises. Depending on the underlying Banach space, we provide necessary and/or sufficient conditions for a function to be integrable. In the last part, the developed theory is applied to define Ornstein-Uhlenbeck processes driven by cylindrical Lévyprocesses and several examples are considered.

1 Introduction

The degree of freedom of models in infinite dimensions is often reflected by the constraint that each mode along a one-dimensional subspace is independently perturbed by the noise. In the Gaussian setting, this leads to the cylindrical Wiener process including from a modeling point of view the very important possibility to describe a Gaussian noise in both time and space with a great flexibility, i.e. space-time white noise. Up to very recently, there has been no analogue for Lévyprocesses. The notion cylindrical Lévyprocess appears the first time in the monograph [26] by Peszat and Zabczyk and it is followed by the works Brze´zniaket al [6], Brze´zniakand Zabzcyk [8], Liu and Zhai [18], Peszat and Zabczyk [27] and Priola and Zabczyk [28]. The first systematic introduction of cylindrical Lévyprocesses appears in our work Applebaum and Riedle [1]. In this work cylindrical Lévyprocesses are introduced as a natural generalisation of cylindrical Wiener processes, and they model a very general, discontinuous noise occurring in the time and state space. The aforementioned literature ([6], [8], [18], [27], [28]) study stochastic evolution equations of the form

dY (t) = AY (t) dt + dL(t) for all t > 0, (1.1) where A is the generator of a strongly continuous semigroup on a Hilbert or Banach space. The driving noise L diﬀers in these publications but it is always constructed in an explicit

∗The author acknowledges the EPSRC grant EP/I036990/1

1 way and it is referred to by the authors as Lévywhite noise, cylindrical stable process or just Lévynoise. The works have in common that the solution of (1.1) is represented by a stochastic convolution integral, which is based either on the one-dimensional integration theory, if the setting allows as for instance in [6], [27], or on moment inequalities for Poisson random measures as for instance in [8]. However, these approaches and the results are tailored to the specific kind of noise under consideration, respectively. The main objective of our work is to develop a general theory of stochastic integration for deterministic integrands, which provides a unified framework for the aforementioned works. Although not part of this work, the results are expected to lead to a better understanding of phenomena, which are individually observed for the solutions of (1.1) in the various models considered in the literature, such as irregularity of trajectories in [6]. In order to be able to develop a general theory, we define cylindrical Lévyprocesses by following the classical approach to cylindrical measures and cylindrical processes, which is presented for example in Badrikian [2] or Schwartz [34]. This systematic approach for cylindrical Lévyprocesses is developed in our work together with Applebaum in [1]. In the current work, we illustrate that those kinds of cylindrical Lévynoise, considered in the literature, are specific examples of a cylindrical Lévyprocess in our approach. Integration for random integrands with respect to other cylindrical processes than the cylindrical Wiener process is only considered in a few works. In fact, we are only aware of two approaches to integration with respect to cylindrical martingales, which originate either in the work developed by Métivierand Pellaumail in [20] and [21] or by Mikuleviˇcius and Rozovskiˇıin [22] and [23]. However, both constructions heavily rely on the assumption of finite weak second moments and are therefore not applicable in our framework. For cylindrical Lévyprocesses with weak second moments, a straightforward integration theory is introduced in Riedle [31]. It is worth mentioning that a localising procedure cannot be applied to cylindrical Lévyprocesses since they do not necessarily attain values in the underlying space. Our work can be seen as a generalisation of the publications by Chojnowska-Michalik [9] on the one hand and by Brze´zniakand van Neerven [7] and by van Neerven and Weis [36] on the other hand. In [9] the author introduces a stochastic integral for deterministic integrands with respect to genuine Lévyprocesses in Hilbert spaces. If we apply our approach to this specific setting, the class of admissible integrands for the integral, developed in our work, is much larger than the one in [9], see Remark 5.11. The articles [7] and [36] introduce a stochastic integral with respect to a cylindrical Wiener processes in Banach spaces. But the approach in [7] and [36] differs from ours as the Gaussian distribution enables the authors to rely on an isometry in terms of square means. Our approach to develop a stochastic integral is based on the idea to introduce first a cylindrical integral, which exists under mild conditions, since cylindrical random variables are more general objects than genuine random variables. A function is then called stochastically integrable if its cylindrical integral is actually induced by a genuine random variable in the underlying Banach space, which is then called the stochastic integral. The advan- tage of this approach is that the latter step is a purely measure-theoretical problem, which can be formulated in terms of the characteristic function of the cylindrical integral. Since the stochastic integral, if it exists, must be infinitely divisible, we can describe the class of integrable functions by using conditions which guarantee the existence of an infinitely divisible random variable for a given semimartingale characteristics. The candidate for the semimartingle characteristics is provided by the cylindrical integral, since its characteristic function coincides with the one of its possible extension to a genuine random variable. Conditions, guaranteeing the existence of an infinitely divisible measure in terms of the characteristic function, are known in many spaces, e.g. in Hilbert spaces and in Banach

2 spaces of Rademacher type. The developed theory of stochastic integration is applied to define Ornstein-Uhlenbeck processes driven by cylindrical Lévyprocesses. We show in two corollaries, that our approach easily recovers results from the literature on the existence of Ornstein-Uhlenbeck processes. The organisation of this paper is as follows. In Section 2 we collect some preliminaries on cylindrical measures and cylindrical random variables, which can be found for example in Badrikian [2] and Schwartz [34]. In Section 3 we recall the definition of a cylindrical Lévy process, based on our paper [1] with Applebaum, and we cite some results on its characteristics from Riedle [30]. Furthermore, some important properties of cylindrical Lévyprocesses are established. Section 4 provides several examples of cylindrical Lévyprocesses. In particular, we show that the noises, considered in the aforementioned publications, are covered by our systematic approach. In Section 5 we develop a theory of stochastic integration for deterministic, operator-valued integrands with respect to cylindrical Lévyprocesses. We finish this work with Section 6 where we apply the developed integration theory to treat Ornstein-Uhlenbeck processes.

2 Preliminaries

Let U be a separable Banach space with dual U ∗. The dual pairing is denoted by hu, u∗i for u ∈ U and u∗ ∈ U ∗. The Borel σ-algebra in U is denoted by B(U) and the closed unit ball at the origin by BU := {u ∈ U : kuk 6 1}. The space of positive, ﬁnite Borel measures on B(U) is denoted by M(U) and it is equipped with the the topology of weak convergence. The Bochner space is denoted by L1([0,T ]; U) and it is equipped with the standard norm. ∗ ∗ ∗ N For every u1, . . . , un ∈ U and n ∈ we deﬁne a linear map

n ∗ ∗ π ∗ ∗ : U → R , π ∗ ∗ (u) = (hu, u i,..., hu, u i). u1 ,...,un u1 ,...,un 1 n Let Γ be a subset of U ∗. Sets of the form

∗ ∗ ∗ ∗ C(u1, . . . , un; B) : = {u ∈ U :(hu, u1i,..., hu, uni) ∈ B} −1 = π ∗ ∗ (B), u1 ,...,un ∗ ∗ Rn where u1, . . . , un ∈ Γ and B ∈ B( ) are called cylindrical sets. The set of all cylindrical sets is denoted by Z(U, Γ) and it is an algebra. The generated σ-algebra is denoted by Zˆ(U, Γ) and it is called the cylindrical σ-algebra with respect to (U, Γ). If Γ = U ∗ we write Z(U) := Z(U, Γ) and Zˆ(U) := Zˆ(U, Γ). A function η : Z(U) → [0, ∞] is called a cylindrical measure on Z(U), if for each finite subset Γ ⊆ U ∗ the restriction of η to the σ-algebra Zˆ(U, Γ) is a measure. A cylindrical measure η is called finite if η(U) < ∞ and a cylindrical probability measure if η(U) = 1. For every function f : U → C which is measurable with respect to Zˆ(U, Γ) for a finite subset Γ ⊆ U ∗ the integral R f(u) η(du) is well defined as a complex valued Lebesgue integral ∗ if it exists. In particular, the characteristic function ϕη : U → C of a finite cylindrical measure η is defined by Z ∗ ihu,u∗i ∗ ∗ ϕη(u ) := e η(du) for all u ∈ U . U Let (Ω, A,P ) be a probability space. The space of equivalence classes of measurable func- 0 tions f :Ω → U is denoted by LP (Ω; U) and it is equipped with the topology of convergence in probability.

3 Similarly to the correspondence between measures and random variables there is an analogous random object associated to cylindrical measures: a cylindrical random variable Z in U is a linear and continuous map

∗ 0 R Z : U → LP (Ω; ).

Here, continuity is with respect to the norm topology on U ∗ and the topology of convergence in probability. A family (Z(t): t > 0) of cylindrical random variables Z(t) is called a cylindrical process. The characteristic function of a cylindrical random variable Z is deﬁned by

∗ ∗ ∗ ϕZ : U → C, ϕZ (u ) = E exp(iZu ) .

∗ ∗ ∗ ∗ ∗ Rn If C = C(u1, . . . , un; B) is a cylindrical set for u1, . . . , un ∈ U and B ∈ B( ) we obtain a cylindrical probability measure η by the prescription

∗ ∗ η(C) := P (Zu1, . . . , Zun) ∈ B .

We call η the cylindrical distribution of Z and the characteristic functions ϕη and ϕZ of η and Z coincide. Conversely, for every cylindrical probability measure η on Z(U) there exist ∗ 0 R a probability space (Ω, A,P ) and a cylindrical random variable Z : U → LP (Ω; ) such that η is the cylindrical distribution of Z, see [35, VI.3.2]. Let ϑ be an inﬁnitely divisible probability measure on B(U). Then the characteristic ∗ ∗ ∗ function ϕϑ : U → C of ϑ is given for each u ∈ U by

Z ∗ ∗ ∗ 1 ∗ ∗ ihu,u i ∗ 1 ϕϑ(u ) = exp ihb, u i − 2 hRu , u i + e − 1 − ihu, u i BU (u) ν(du) , (2.1) U where b ∈ U, R: U ∗ → U is the covariance operator of a Gaussian measure on B(U) and ν is a σ-finite measure on B(U). Since the triplet (b, R, ν) is unique ([17, Th.5.7.3]), it characterises the distribution of the probability measure ϑ, and it is called the characteristics of ϑ. If X is an U-valued random variable which is infinitely divisible, then we call the characteristics of its probability distribution the characteristics of X. In general Banach spaces it is not as straightforward to define a Lévymeasure as in Hilbert spaces. In this work we use the following result (Theorem 5.4.8 in Linde [12]) as the definition: a σ-finite measure ν on a Banach space U is called a Lévymeasure if Z (i) hu, u∗i2 ∧ 1 ν(du) < ∞ for all u∗ ∈ U ∗; U (ii) there exists a measure on B(U) with characteristic function

Z ∗ ∗ ihu,u i ∗ 1 ϕ(u ) = exp e − 1 − ihu, u i BU (u) ν(du) . U

Let {F t}t>0 be a filtration for the probability space (Ω, A,P ). An adapted, stochastic process L := (L(t): t > 0) with values in U is called a Lévyprocess if L(0) = 0 P -a.s., L has independent and stationary increments and L is continuous in probability. It follows that there exists a version of L with paths which are continuous from the right and have limits from the left (càdlàgpaths). The random variable L(1) is infinitely divisible and we call its characteristics the characteristics of L.

4 3 Cylindrical L´evyprocesses

Let U be a separable Banach space. A cylindrical probability measure η on Z(U) is called inﬁnitely divisible if for each k ∈ N there exists a cylindrical probability measure ηk such ∗k ∗ that η = ηk) . In [1] and [30] we show that the characteristic function ϕη : U → C of η can be represented by

Z ∗ ∗ ∗ 1 ∗ ihu,u i ∗ 1 ∗ ϕη(u ) = exp ia(u ) − 2 qu + e − 1 − ihu, u i BR (hu, u i) µ(du) U (3.1) =: exp Ψ(u∗), where a: U ∗ → R is a mapping with a(0) = 0 and which is continuous on ﬁnite dimensional subspaces, q : U ∗ → R is a quadratic form and µ is a cylindrical measure on Z(U) satisfying

Z hu, u∗i2 ∧ 1 µ(du) < ∞ for all u∗ ∈ U ∗. (3.2) U Consequently, the triplet (a, q, µ) characterises the distribution of the cylindrical measure η and thus, it is called the (cylindrical) characteristics of η. The mapping Ψ: U ∗ → C is called the (cylindrical) symbol of η. We call a cylindrical measure µ on Z(U) a (cylindrical) Lévymeasure if it satisfies (3.2). However, note that it is not sufficient for a cylindrical measure µ to satisfy (3.2) in order to guarantee that there exists a corresponding infinitely divisible cylindrical measure with characteristics (0, 0, µ), see [30]. For a cylindrical or classical Lévymeasure µ we denote µ−(C) := µ(−C) for all C ∈ Z(U). A cylindrical process (L(t): t > 0) is called a cylindrical Lévyprocess in U if for all ∗ ∗ ∗ N u1, . . . , un ∈ U and n ∈ we have that

∗ ∗ (L(t)u1,...,L(t)un): t > 0 is a Lévyprocess in Rn. This definition is introduced in our work [1]. It follows that the cylindrical distribution η of L(1) is infinitely divisible and that the characteristic function of L(t) for all t > 0 is given by ∗ ∗ ∗ ϕL(t) : U → C, ϕL(t)(u ) = exp tΨ(u ) , (3.3) where Ψ: U ∗ → C is the symbol of η. We call the symbol Ψ and the characteristics (a, q, µ) of η the (cylindrical) symbol and the (cylindrical) characteristics of L. A cylindrical Lévyprocess (L(t): t > 0) with characteristics (a, q, µ) can be decomposed into

L(t) = W (t) + P (t) for all t > 0, (3.4) ∗ 0 R where W (t) and P (t) are linear maps from U to LP (Ω; ), see [1, Th.3.9]. For each ∗ ∗ ∗ ∗ u ∈ U the stochastic processes (W (t)u : t > 0) and (P (t)u : t > 0) are unique up to indistinguishability by [15, Th.I.4.18]. In addition to the assumed continuity of the operator L(t) we require in this work that W (t) and P (t) are continuous for each t > 0, in which case (W (t): t > 0) is a cylindrical L´evyprocess with characteristics (0, q, 0) and (P (t): t > 0) is an independent, cylindrical L´evyprocess with characteristics (a, 0, µ). Lemma 4.4. in [30] guarantees that the characteristics (a, q, µ) obeys: (1) a: U ∗ → R is continuous;

5 (2) there exists a positive, symmetric operator Q: U ∗ → U ∗∗ such that

qu∗ = hu∗, Qu∗i for all u∗ ∈ U ∗;

∗ ∗ ∗ ∗ ∗ ∗ (3) for every sequence (un)n∈N ⊆ U with kun − u0k → 0 for some u0 ∈ U it follows that 2 ∗ −1 2 ∗ −1 R |β| ∧ 1 µ ◦ (un) (dβ) → |β| ∧ 1 µ ◦ (u0) (dβ) weakly in M( ). (3.5)

In this case, we replace the covariance q by the covariance operator Q and write (a, Q, µ) for the cylindrical characteristics. We will need several times the following property of an arbitrary cylindrical Lévymeasure in a Banach space. For classical Lévymeasures, the same property can be deduced by different arguments, see [17, Pro.5.4.5]. Lemma 3.1. Let µ be the cylindrical Lévymeasure of a cylindrical Lévyprocess in U. Then for every ε > 0 there exists a δ > 0 such that Z ∗ 2 sup |hu, u i| ∧ 1 µ(du) 6 ε. ∗ ku k6δ U Proof. Due to (3.4) we can assume that the cylindrical Lévyprocess L has the characteristics (a, 0, µ). If L0 denotes an independent copy of L then the cylindrical Lévyprocess L˜ := L−L0 has the characteristics (0, 0, µ + µ−). ∗ ∗ ∗ ∗ Define for every u ∈ U the cylindrical set D(u ) := {u ∈ U : |hu, u i| 6 1}. The 1 2 − inequality 1 − cos(β) > 3 β for all |β| 6 1 implies by using the symmetry of µ + µ , that the characteristic function of L˜(1) satisfies for each u∗ ∈ U ∗: Z ∗ ∗ − ϕL˜(1)(u ) = exp − 1 − cos(hu, u i) (µ + µ )(du) U Z ! ∗ − 6 exp − 1 − cos(hu, u i) (µ + µ )(du) D(u∗) Z ! 2 ∗ 2 6 exp − 3 |hu, u i| µ(du) . D(u∗)

Consequently, we obtain Z ∗ 2 3 ∗ ∗ ∗ |hu, u i| µ(du) 6 − 2 ln(ϕL˜(1)(u )) for all u ∈ U . D(u∗)

˜ ∗ 0 R ∗ R Since L(1): U → LP (Ω; ) is continuous, its characteristic function ϕL˜(1) : U → is continuous, see [35, Pro.IV.3.4]. Therefore, there exists a δ1 > 0 such that Z sup |hu, u∗i|2 µ(du) < ε. (3.6) ∗ ∗ ku k6δ1 D(u ) For the second part of the proof, we deﬁne d(u∗) := µD(u∗)c for all u∗ ∈ U ∗, and we show that for every ε > 0 there exists a δ2 > 0 such that

∗ sup d(u ) 6 ε. (3.7) ∗ ku k6δ2

6 ∗ Assume for a contradiction that (3.7) is not satisﬁed. Then there exists a sequence (un)n∈N ⊆ ∗ ∗ ∗ N N U with un → 0 as n → ∞ and d(un) > ε for all n ∈ . For each n ∈ deﬁne the stopping ∗ time τ ∗ := inf{t 0 : |(L(t) − L(t−))u | > 1}. Since for each n ∈ N the stopping time un > n ∗ τ ∗ is exponentially distributed with parameter d(u ) it follows that un n

∗ 1 −d(u∗ )T −εT P sup |L(t)u | P τ ∗ > T = e n < e for all n ∈ N. (3.8) n 6 2 6 un t∈[0,T ] Let D([0,T ]; R) denote the space of functions on [0,T ] with càdlàgtrajectories and endow this space with the supremum norm. Define the mapping L: U ∗ → L0(Ω; D([0,T ]; R)) by Lu∗ := (L(t)u∗ : t ∈ [0,T ]). It follows by the closed graph theorem for F -spaces (see [37, ∗ Th.II.6.1]), that L is a continuous mapping. Consequently, we obtain supt∈[0,T ] L(t)un → 0 in probability as n → ∞, which contradicts (3.8). Thus, we have proved equality (3.7), which together with (3.6) completes the proof. The first application of the previous Lemma 3.1 establishes that the cylindrical symbol Ψ maps bounded sets into bounded sets. Lemma 3.2. The cylindrical symbol Ψ satisfies for each c > 0:

sup |Ψ(u∗)| < ∞. ∗ ku k6c Proof. Let L be a cylindrical L´evy process with symbol Ψ and characteristics (a, Q, µ), so that Ψ is of the form (3.1). Since L(1)(βu∗) and βL(1)u∗ are identically distributed for every u∗ ∈ U ∗ and β > 0, equating their L´evy-Khintchine formula yields Z ∗ ∗ ∗ 1 ∗ 1 ∗ a(βu ) = βa(u ) + β hu, u i BR (βhu, u i) − BR (hu, u i) µ(du). (3.9) U The second term on the right hand side can be estimated by Z |hu, u∗i| 1 (βhu, u∗i) − 1 (hu, u∗i) µ(du) BR BR U Z Z Z ∗ 2 ∗ 2 1 6 hu, u i µ(du) + µ(du) = hu, u i ∧ β2 µ(du). (3.10) ∗ 1 ∗ 1 |hu,u i|6 β |hu,u i|> β U

∗ The continuity of a and a(0) = 0 imply that there exists a δ > 0 such that |a(u )| 6 1 for ∗ c all ku k 6 δ. By choosing β = δ it follows from (3.9) and (3.10) by Lemma 3.1, that

∗ Z ∗ u ∗ 2 1 sup |a(u )| 6 β sup a β + β sup |hu, u i| ∧ β2 µ(du) < ∞. (3.11) ∗ ∗ ∗ ku k6c ku k6c ku k6c U Boundedness of the term in (3.1) involving Q can easily be established since Q ∈ L(U ∗,U ∗∗). Applying the estimate

Z ∗ Z ihu,u i ∗ 1 ∗ ∗ 2 sup e − 1 − ihu, u i BR (hu, u i) µ(du) 6 2 sup |hu, u i| ∧ 1 µ(du) ∗ ∗ ku k6c U ku k6c U completes the proof by another application of Lemma 3.1. It is well known that if the covariance of a Gaussian cylindrical measure is majorised by the covariance of a Gaussian measure, it extends to a measure and is Gaussian. Next we will derive the analogue result for cylindrical L´evymeasures, following the presentation of the result in the Gaussian setting:

7 Theorem 3.3. Let η be a centralised, Gaussian cylindrical measure on Z(U) with covariance q : U ∗ → R, i.e. Z q(u∗) = hu, u∗i2 η(du). U If ϑ is a Gaussian measure on B(U) with covariance operator R: U ∗ → U satisfying ∗ ∗ ∗ ∗ ∗ q(u ) 6 hu , Ru i for all u ∈ U , then η extends to a measure on B(U) and the extension is Gaussian. Proof. See Theorem 3.3.1 in [3]. We extend this result to cylindrical Lévymeasures by generalising Prokhorov’s theorem on projective limits ([4, Th.9.12.2] or [35, Th.VI.3.2]) to σ-finite measures. We follow here the proof of [14] in the form as nicely rewritten in [3]. Theorem 3.4. Let µ be a cylindrical Lévymeasure on Z(U) and ν be a Lévymeasure on B(U) satisfying µ 6 ν on Z(U). Then µ extends to a σ-finite measure on B(U) and the extension is a Lévymeasure. Proof. Fix a δ > 0 and define for each d ∈ N the rectangle n o d Rd Rδ := (β1, . . . , βd) ∈ : sup |βi| 6 δ , i=1,...,d and Bδ := {u ∈ U : kuk 6 δ}. Since U is separable we can choose a norming sequence ∗ ∗ ∗ {uk}k∈N ⊆ U with kukk = 1, i.e. ∗ kuk = sup |hu, uki| for all u ∈ U. k∈N N Rk ∗ ∗ We define for every k ∈ the mapping πk : U → by πk(u) := hu, u1i,..., hu, uki , and the finite measure k −1 k k µ := µ ◦ π |Rk k on B(R \R ). δ k \Rδ δ

Since ν is a L´evymeasure there exists an increasing sequence {Kn}n∈N of compact sets Kn ⊆ U (see [17, Th.5.4.8]), such that 1 ν u ∈ U : u ∈ Kc , kuk > δ for all n ∈ N . n 6 n c By denoting the constant cδ := ν(Bδ ) deﬁne the set n 1 o S := ϑ ∈ M(U): ϑ(U) c , ϑ(Kc ) for all n ∈ N . δ 6 δ n 6 n

Obviously, the set Sδ is non-empty and relatively compact in M(U) by Prohorov’s Theorem; see [35, Th.I.3.6]. Furthermore, for each k, n ∈ N we obtain k c −1 c k µδ πk(Kn) = µ ◦ πk (πk(Kn)) \Rδ −1 c k 6 ν ◦ πk (πk(Kn)) \Rδ c ∗ = ν u ∈ U : u ∈ Kn, sup |hu, ui i| > δ i=1,...,k c 6 ν {u ∈ U : u ∈ Kn, kuk > δ} 1 . 6 n

8 N k Theorem 9.1.9 in [4] implies that for each k ∈ there exists a measure ϑδ ∈ M(U) such that k −1 k Rk k ϑδ ◦ πk = µδ on B( \Rδ ), k k Rk k k c 1 N k Rk k with ϑδ (U) = µδ ( \Rδ ) and ϑδ (Kn) 6 n for all n ∈ . Since µδ ( \Rδ ) 6 cδ, the set k ¯ −1 k Rk k Sδ := ϑ ∈ Sδ : ϑ ◦ πk = µδ on B( \Rδ ) R` Rk is non-empty. For k 6 ` denote by πk` the natural projection from to . Since −1 Rk k R` ` ` πk` ( \Rδ ) ⊆ \Rδ we obtain for ϑ ∈ Sδ that −1 −1 −1 ` −1 −1 −1 −1 Rk k ϑ ◦ πk = (ϑ ◦ π` ) ◦ πk` = µδ ◦ πk` = (µ ◦ π` ) ◦ πk` = µ ◦ πk on B( \Rδ ), ` k ¯ k N which shows Sδ ⊆ Sδ . Since Sδ is compact and Sδ is closed for all k ∈ , the nested system k N {Sδ : k ∈ } has a non-empty intersection. Thus, for each δ > 0 there exists a measure ϑδ ∈ M(U) satisfying −1 −1 Rk k N ϑδ ◦ πk = µ ◦ πk on B( \Rδ ) for all k ∈ . c The measure ϑδ is uniquely determined on B(U) ∩ Bδ since the family of sets −1 Rk k Rk N {Z ∈ Z(U): Z = πk (B ∩ \Rδ ) for B ∈ B( ), k ∈ } c generates the σ-algebra B(U) ∩ Bδ and it is closed under intersection. Define the measure ˜ 1 1 ˜ ϑ1/k to be the restriction of ϑ1/k on the disc {u ∈ U : k < kuk 6 k−1 } for k > 2 and ϑ1 to c be the restriction of ϑ1 to B1. Then ∞ X ˜ ϑ := ϑ1/k k=1 defines a σ-finite measure ϑ on B(U) satisfying ϑ = µ on Z(U). Since ϑ 6 ν on Z(U) and Z(U) is a generator of B(U) closed under intersection, we have ϑ 6 ν on B(U). Proposition 5.4.5 in [17] guarantees that ϑ is a Lévymeasure.

4 Examples of cylindrical L´evyprocesses

Example 4.1. If (Y (t): t > 0) is a genuine Lévyprocess with values in a Banach space U, then, for t > 0, ∗ 0 R ∗ ∗ L(t): U → LP (Ω; ),L(t)u = hY (t), u i defines a cylindrical Lévyprocess in U. If (b, R, ν) is the characteristics of Y , then the cylindrical characteristics (a, Q, µ) of L is given by Z ∗ ∗ ∗ 1 ∗ 1 a(u ) = hb, u i + hu, u i BR (hu, u i) − BU (u) ν(du),Q = R, µ = ν. U The existence of the integral is derived in Lemma 5.7. The asymmetry of the classical characteristics and the cylindrical characteristics of Y is due to the fact, that in the cylindrical perspective the entry µ is only a cylindrical measure 1 and therefore, the truncation function u 7→ BU (u) cannot be integrated with respect to µ. A more illustrative reason is that a classical Lévyprocess obviously has jumps in the underlying Banach space U, whereas it is not clear in which space the jumps of a cylindrical Lévyprocess occur.

9 An appealing way to construct a cylindrical L´evyprocess is by a series of real valued L´evyprocesses. We denote here by `p(R) for p ∈ [1, ∞] the spaces of real valued sequences.

Lemma 4.2. Let U be a Hilbert space with an orthonormal basis (ek)k∈N and let (`k)k∈N be a sequence of independent, real valued L´evyprocesses with characteristics (bk, rk, νk) for k ∈ N. Then the following are equivalent:

2 (a) For each (αk)k∈N ∈ ` (R) we have

∞ Z X 1 (i) BR (αk) |αk| bk + β νk(dβ) < ∞; −1 k=1 1<|β|6|αk| ∞ (ii) (rk)k∈N ∈ ` (R); ∞ Z X 2 (iii) |αkβ| ∧ 1 νk(dβ) < ∞. R k=1 ∗ ∗ (b) For each t > 0 and u ∈ U the sum

∞ ∗ X ∗ L(t)u := hek, u i`k(t) (4.1) k=1 converges P -a.s. N If in this case the set {ϕ`k(1) : k ∈ } is equicontinuous at 0, then (L(t): t > 0) deﬁnes a cylindrical L´evyprocess in U with cylindrical characteristics (a, Q, µ) obeying

∞ Z ∗ X ∗ 1 ∗ 1 a(u ) = hek, u i bk + β BR (hek, u iβ) − BR (β) νk(dβ) , R k=1 ∞ ∞ ∗ X ∗ ∗ −1 X ∗ −1 Qu = hek, u irkek, µ ◦ (u ) (dβ) = νk ◦ mk(u ) (dβ), k=1 k=1

∗ ∗ ∗ ∗ ∗ for each u ∈ U , where mk(u ): R → R is deﬁned by mk(u )(β) = hek, u iβ.

Proof. Deﬁne for an arbitrary sequence (αk)k∈N ⊆ R and n ∈ N the partial sum

n X Sn(t) := αk`k(t) for all t > 0. (4.2) k=1

It follows by using [33, Pro.11.10], that (Sn(t): t > 0) is a L´evyprocess with characteristics

n n n X X X b(n) := b0 , r(n) := α2r , ν(n)(dβ) := ν ◦ m−1 (dβ), k k k k αk k=1 k=1 k=1 R R 0 where mαk : → , mαk (β) = αkβ and the reals bk are deﬁned by Z 0 1 1 bk := αkbk + αkβ BR (αkβ) − BR (β) νk(dβ). R

∗ ∗ ∗ For establishing the implication (a) ⇒ (b) ﬁx u ∈ U and set αk := hek, u i. Due (n) to Condition (i) there exists b ∈ R such that limn→∞ b = b. Conditions (ii) and (iii)

10 guarantee that for every continuous and bounded function f : R → R we have Z (n) 2 (n) f(β) r δ0(dβ) + |β| ∧ 1 ν (dβ) R n n Z X 2 X 2 = f(0) αkrk + f(αkβ) |αkβ| ∧ 1 νk(dβ) R k=1 k=1 ∞ ∞ Z X 2 X 2 → f(0) αkrk + f(αkβ) |αkβ| ∧ 1 νk(dβ) as n → ∞. R k=1 k=1

Theorem VII.2.9 and Remark VII.2.10 in [15] imply that the sum Sn(t) converges weakly ∗ and therefore P -a.s. to an inﬁnitely divisible random variable L(t)u for each t > 0. Conversely, it follows from (b) that the sum Sn(1), deﬁned in (4.2), converges weakly for 2 (n) every (αk)k∈N ∈ ` (R). Theorem VII.2.9 in [15] implies that b converges as n → ∞, i.e.

∞ Z X 1 1 αk bk + β BR (αkβ) − BR (β) νk(dβ) < ∞ R k=1

2 for every (αk)k∈N ∈ ` (R). Since for each k ∈ N the term in the bracket does not depend on the sign of αk we can choose αk such that each summand is positive and we obtain ∞ Z X 1 1 |αk| bk + β BR (αkβ) − BR (β) νk(dβ) < ∞, R k=1 which yields Condition (i) as |αk| 6 1 for suﬃciently large k. Remark VII.2.10 in [15] implies that for every continuous, bounded function f : R → R the sum n Z X 2 2 αkrkf(0) + f(αkβ) |αkβ| ∧ 1 νk(dβ) R k=1 N converges as n → ∞. Since we can assume that rk > 0 for every k ∈ , Conditions (ii) and (iii) are implied by choosing f(·) = 1, which completes the proof of the equivalence (a) ⇔ (b). ∗ 0 R ∗ ∗ Clearly, L(t): U → LP (Ω; ) is linear. If a sequence (un)n∈N ⊆ U converges to 0 then ∗ N N hek, uni → 0 as n → ∞ uniformly in k ∈ . The equicontinuity of {ϕ`k(1) : k ∈ } implies ∗ N for each t > 0 that ϕ`k(t)(hek, uni) → 1 for n → ∞ uniformly in k ∈ . Thus, we obtain

∞ ∞ ∗ Y ∗ Y ∗ lim ϕL(t)(un) = lim ϕ` (t)(hek, uni) = lim ϕ` (t)(hek, uni) = 1, n→∞ n→∞ k n→∞ k k=1 k=1 which shows the continuity of L(t). Finally, L has weakly independent increments, that is the random variables

∗ ∗ L(t1) − L(t0) u1,..., L(tn) − L(tn−1) un ∗ ∗ ∗ N ∗ are independent for every 0 6 t0 6 ··· 6 tn, u1, . . . , un ∈ U and n ∈ . Since (L(t)u : ∗ ∗ t > 0) is a real valued L´evyprocess for each u ∈ U it follows from [1, Le.3.8] that L is a cylindrical L´evyprocess in U.

11 P The convergence (4.1) is called the weakly P -a.s. convergence of the sum ek`k(t) for each t > 0. If there exists a random variable Y (t):Ω → U for each t > 0 such that

∞ X Y (t) = ek`k(t) P -a.s. in U, k=1 then the sum is called strongly P -a.s.convergent. Obviously, in this case, we have L(t)u∗ = ∗ ∗ ∗ hY (t), u i for every u ∈ U and t > 0, and (Y (t): t > 0) is a U-valued L´evyprocess. One can easily show a similar result to Lemma 4.2 but we skip this; a special case can be found in [26, Th.4.13]

Example 4.3. Let `k be defined by `k(·) := σkwk(·) where (σk)k∈N ⊆ R and (wk)k∈N is a sequence of independent, real valued standard Brownian motions. Then the series in (4.1) ∞ defines a cylindrical Lévyprocess L if and only if (σk)k∈N ∈ ` . In this case L is called a cylindrical Wiener process. Its covariance operator Q is given by

∞ ∗ ∗ X 2 ∗ Q: U → U, Qu = σkhek, u iek. k=1 This deﬁnition of a cylindrical Wiener process is consistent with other deﬁnitions which can be found in the literature, see [29].

Example 4.4. For a sequence (hk)k∈N of independent, real valued Poisson processes with intensity 1 and a sequence σ := (σk)k∈N ⊆ R we define `k(·) := σkhk(·). In this case, the sum (4.1) defines a cylindrical Lévyprocess if and only if σ ∈ `2. The sum converges strongly 1 if and only if σ ∈ ` . If (hk)k∈N is a sequence of independent, real valued compensated Poisson processes with intensity 1, then the sum converges weakly if and only if σ ∈ `∞ and strongly if and only if σ ∈ `2.

Example 4.5. Let (hk)k∈N be a family of independent, identically distributed, real valued, standardised, symmetric, α-stable Lévyprocesses hk for α ∈ (0, 2). Then the character- 1 −1−α istics of hk is given by (0, 0, ρ) with Lévymeasure ρ(dβ) = 2 |β| dβ. For a sequence σ := (σk)k∈N ⊆ R define for each k ∈ N the Lévyprocess `k(·) := σkhk(·). Then the characteristics of `k is given by (0, 0, νk) with

ν := ρ ◦ m−1, (4.3) k σk where mα : R → R is defined by mαβ = αβ for some α ∈ R. With this choice of (`k)k∈N it follows by Lemma 4.2 that the sum (4.1) defines a cylindrical Lévyprocess if and only if

∞ Z ∞ X 2 2 X α |αkβ| ∧ 1 νk(dβ) = α(2−α) |αkσk| < ∞ R k=1 k=1

2 (2α)/(2−α) for every (αk)k∈N ∈ ` , which is equivalent to σ ∈ ` . The cylindrical Lévyprocess is U-valued, i.e. the sum converges strongly, if and only if σ ∈ `α. Remark 4.6. The publication [28] treats the cylindrical Lévyprocess introduced in Exam- ple 4.5 and it is called cylindrical stable noise. This specific example of a cylindrical Lévy process appears also in the publications [6], [18] and [27]. However, since the authors do not follow the cylindrical approach, they do not require that the sum (4.1) is finite, i.e. they do not impose any conditions on the sequence σ. Although this is more general, it does not

12 match the usual framework, if one understands cylindrical Lévyprocesses as a generalisation of cylindrical Wiener processes. All the different definitions of cylindrical Wiener processes, one can find in the literature, have in common that the corresponding sum of the form (4.1) converges weakly as in Example 4.3. Example 4.7. In [8], the authors construct a noise by subordination which they call Lévy white noise. In the following result we define this noise in our setting and derive its cylindrical characteristics. In contrast to the original source, our cylindrical approach enables us to introduce this noise without referring to any other space than the underlying Banach space U, which we consider to be more natural. Lemma 4.8. If W is a cylindrical Wiener process in U with covariance operator C and ` is an independent, real valued subordinator with characteristics (α, 0, ρ) then ∗ ∗ ∗ ∗ L(t)u := W (`(t))u for all u ∈ U , t > 0, (4.4) defines a cylindrical Lévyprocess (L(t): t > 0) with characteristics (0, Q, µ) given by Q = αC, µ = (γ ⊗ ρ) ◦ κ−1, where γ is the canonical Gaussian cylindrical measure on the reproducing kernel Hilbert space HC of C with embedding iC : HC → U and √ κ: HC × R+ → U, κ(h, s) := s iC h. Proof. The very definition of L implies by Lemma 3.8 in [1] using independence of W and ` that L is a cylindrical Lévyprocess. The characteristic function of the subordinator ` can be analytically continued, such that for each t > 0 we obtain the Laplace transform of `(t) by E [exp(−β`(t))] = exp (−tτ(β)) for all β > 0, where the Laplace exponent τ is defined by Z ∞ τ(β) := αβ + 1 − e−βs ρ(ds) for all β > 0, 0 ∗ ∗ see [33, Th.24.11]. Independence of W and ` implies that for each t > 0 and u ∈ U the characteristic function ϕL(t) of L(t) is given by Z ∞ ∗ h iW (s)u∗ i ϕL(t)(u ) = E e P`(t)(ds) 0 ∞ Z 1 ∗ ∗ − 2 shCu ,u i 1 ∗ ∗ = e P`(t)(ds) = exp −tτ 2 hCu , u i . (4.5) 0 ∗ By using C = iC iC , γ(HC ) = 1 and the symmetry of the canonical Gaussian cylindrical measure γ we obtain ∞ Z 1 ∗ ∗ e− 2 shCu ,u i − 1 ρ(ds) 0 ∞ Z Z √ ∗ = ei s hiC h,u i − 1 γ(dh) ρ(ds) 0 HC ∞ Z Z √ ∗ √ √ i s hiC h,u i ∗ 1 ∗ = e − 1 − i s hiC h, u i BR ( s hiC h, u i) γ(dh) ρ(ds) 0 HC Z ∗ ihu,u i ∗ 1 ∗ −1 = e − 1 − ihu, u i BR (hu, u i) (γ ⊗ ρ) ◦ κ (du). (4.6) U

13 Note that (γ ⊗ ρ) ◦ κ−1 is a cylindrical Lévymeasure since for each u∗ ∈ U ∗ we have Z Z ∞ Z ∗ 2 −1 ∗ 2 hu, u i ∧ 1 (γ ⊗ ρ) ◦ κ (du) = shiC h, u i ∧ 1 γ(dh) ρ(ds) U 0 HC Z ∞ Z ∗ 2 6 s hiC h, u i γ(dh) ∧ 1 ρ(ds) 0 HC Z ∞ = shCu∗, u∗i ∧ 1 ρ(ds) < ∞. 0 The finiteness of the last integral is shown in [33, Th.21.5]. Applying equality (4.6) to the representation (4.5) yields that the characteristic function of L(t) is of the claimed form. The previous example highlights an important difference between cylindrical Wiener and cylindrical Lévyprocesses. According to the Karhunen-Loève expansion, each cylindrical Wiener processes can be represented by a sum of independent U-valued random variables, see for example [29, Th.20]. However, such kind of representation cannot be expected for the noise constructed in Lemma 4.8. Example 4.9. Another example of a cylindrical Lévyprocess is the impulsive cylindrical 2 R process on Lλ(O; ), which is introduced in the monograph [26]. In our work [1] we show that also this kind of a noise can be understood as a specific example of a cylindrical Lévy approach in our general approach.

5 Stochastic integration

In this section, U and V are separable Banach spaces and (L(t): t > 0) denotes a cylindrical Lévyprocess in U with characteristics (a, Q, µ). Let f : [0,T ] → L(U, V ) be a deterministic function, where L(U, V ) denotes the space of linear, bounded functions from U to V . The aim of this section is to define a stochastic integral Z IA := f(s) dL(s) A as a V -valued random variable for each Borel set A ⊆ [0,T ]. Our approach is based on the idea that the random variable IA, if it exists, must be infinitely divisible and thus, its probability distribution is uniquely described by its characteristics, say (bA,RA, νA). If we have a candidate for the characteristics of IA and if there are conditions known (such as in Hilbert spaces) guaranteeing the existence of an infinitely divisible random variable in terms of its prospective characteristics, then we can describe the class of integrable functions f : [0,T ] → L(U, V ). This approach works in every separable Banach space V in which explicit conditions on the characteristics are known guaranteeing the existence of a corresponding infinitely divisible measure. In order to have a candidate for the characteristics of IA at hand for formulating the con- ∗ 0 R ditions on integrability, we first introduce a cylindrical random variable ZA : V → LP (Ω; ) as a cylindrical integral of f. Then we call f integrable with respect to L if for each Borel set A ⊆ [0,T ] the cylindrical integral ZA is induced by a classical random variable IA :Ω → V , i.e.

∗ ∗ ∗ ∗ ZAv = hIA, v i for all v ∈ V .

In this way one can think of IA as a stochastic Pettis integral.

14 For a well defined integral we expect that Z Z h f(s) dL(s), v∗i = f ∗(s)v∗ dL(s) for all v∗ ∈ V ∗. A A Thus, in a first step, we introduce a real valued stochastic integral for U ∗-valued functions with respect to a cylindrical Lévyprocess L. For this purpose, we initially consider simple U ∗-valued functions. A deterministic function g : [0,T ] → U ∗ is called simple if there is a partition 0 = t0 6 t1 6 ··· 6 tm = T such that g is constant on the open interval (tk, tk+1) for each k = 0, . . . , m − 1. The space of all simple functions is denoted by S([0,T ]; U ∗) and it is endowed with the supremum norm

kgk∞ := sup kg(s)k . s∈[0,T ]

Let G([0,T ]; U ∗) denote the space of deterministic regulated functions; these are all functions g : [0,T ] → U ∗ such that for every t ∈ (0,T ) there exists the limit of g on the left and on the right of t and on the right of 0 and on the left of T . In other words, each regulated function has only discontinuities of the ﬁrst kind. It is shown in [5, Ch.II.1.3] or [11, Ch.VII.6] that a function g is regulated if and only if it can be uniformly approximated by step functions. In particular, regulated functions are bounded and the space G([0,T ]; U ∗) can be equipped with the supremum norm, which turns it into a Banach space. ∗ ∗ For a simple function g ∈ S([0,T ]; U ) which attains the value uk on the interval (tk, tk+1) ∗ 0 R for k = 0, . . . , m − 1 deﬁne a mapping J : S([0,T ]; U ) → LP (Ω; ) by

m−1 X ∗ J(g) := L(tk+1) − L(tk) (uk). (5.1) k=0 In order to show that the mapping J is continuous we need the following result. Lemma 5.1. If Ψ: U ∗ → C is the symbol of a cylindrical L´evyprocess L then the mapping

G([0,T ]; U ∗) → L1([0,T ]; C), g 7→ Ψ(g(·)) is continuous. Proof. Continuity of Ψ and Lemma 3.2 guarantee that Ψ(g(·)) ∈ L1([0,T ]; C) for g ∈ ∗ ∗ ∗ G([0,T ]; U ). Let (gn)n∈N be a sequence in G([0,T ]; U ) converging to g ∈ G([0,T ]; U ). Recall that (a, Q, µ) denotes the cylindrical characteristics of L and Ψ is given in (3.1). Inequality (3.11) shows that a: U ∗ → R maps bounded sets into bounded sets and thus, Lebesgue’s dominated convergence theorem implies Z T lim |a(gn(s)) − a(g(s))| ds = 0. n→∞ 0 Another application of Lebesgue’s dominated convergence theorem shows Z T lim |hgn(s), Qgn(s)i − hg(s), Qg(s)i| ds = 0. n→∞ 0 Deﬁne for each n ∈ N the function Z C ihu,gn(s)i 1 hn : [0,T ] → , hn(s) = e − 1 − ihu, gn(s)i BR (hu, gn(s)i) µ(du), U

15 and the function ( iβ 1 e −1−iβ BR (β) , if β 6= 0, f : R → C, f(β) = β2∧1 1 − 2 , if β = 0. Clearly, the function f is bounded and continuous. Lemma 3.1 guarantees Z 2 sup sup |hn(s)| = sup sup f(hu, gn(s)i) hu, gn(s)i ∧ 1 µ(du) n∈N s∈[0,T ] n∈N s∈[0,T ] U Z 2 6 kfk∞ sup sup hu, gn(s)i ∧ 1 µ(du) n∈N s∈[0,T ] U < ∞.

Since gn(s) → g(s) for each s ∈ [0,T ], it follows from (3.5): Z 2 −1 lim hn(s) = lim f(β) |β| ∧ 1 µ ◦ gn(s) )(dβ) n→∞ n→∞ R Z = f(β) |β|2 ∧ 1 µ ◦ g(s)−1)(dβ) R =: h(s). Lebesgue’s dominated convergence theorem implies Z T lim |hn(s) − h(s)| ds = 0, n→∞ 0 which completes the proof. ∗ 0 R Lemma 5.2. The operator J : S([0,T ]; U ) → LP (Ω; ) deﬁned in (5.1) is continuous where 0 R LP (Ω; ) is equipped with the topology of convergence in probability. ∗ ∗ Proof. Let (gn)n∈N ⊆ S([0,T ]; U ) be a sequence converging to g in S([0,T ]; U ). Then, by linearity of J it follows that J(gn) converges to J(g) in probability if and only if J(gn−g) → 0 in probability. However, the latter convergence occurs if and only if J(gn − g) → 0 weakly. Independent increments of L yields that the characteristic function of J(gn) is given by Z T ! R C ϕJ(gn) : → , ϕJ(gn)(β) = exp Ψ(βgn(s)) ds . 0 R Consequently, it follows from Lemma 5.1 that ϕJ(gn)(β) converges to ϕJ(g)(β) for all β ∈ , which completes the proof. ∗ 0 R The mapping J : S([0,T ]; U ) → LP (Ω; ) is linear and uniformly continuous, since the 0 R metric of LP (Ω; ) is translation invariant. The principle of extension by continuity ([12, Th.I.6.17]) enables us to extend the mapping J to the space G([0,T ]; U ∗), i.e. we deﬁne 0 R J(g) := lim J(gn) in LP (Ω; ), n→∞ ∗ ∗ where (gn)n∈N ⊆ S([0,T ]; U ) is chosen such that gn → g in G([0,T ]; U ). Lemma 5.1 implies that the characteristic function of J(g) is given by

Z T ! ϕJ(g) : R → C, ϕJ(g)(β) = exp Ψ(βg(s)) ds . (5.2) 0

16 Now we come back to the original aim to introduce a stochastic integral for integrands with values in L(U, V ) as a genuine V -valued random variable. For that purpose, let f : [0,T ] → L(U, V ) be a function and for each s ∈ [0,T ] denote by f ∗(s) the adjoint of f(s): U → V . We say that the function f : [0,T ] → L(U, V ) is weakly in G([0,T ]; U ∗) if f ∗(·)v∗ is in G([0,T ]; U ∗) for each v∗ ∈ V ∗. Clearly, the function s 7→ f ∗(s)v∗ is measurable for each v∗ ∈ V ∗. Lemma 5.3. If τ : [0,T ] → R is measurable and bounded and g ∈ G([0,T ]; U ∗) then τ(·)g(·) ∈ G([0,T ]; U ∗).

Proof. Since τ is bounded there exist simple functions τn converging uniformly to τ. If ∗ (gn)n∈N ⊆ S([0,T ]; U ) converges uniformly to g it follows

kτg − τngnk∞ 6 kτk∞ kg − gnk∞ + kτ − τnk∞ kgnk∞ → 0. Thus, τg can be uniformly approximated by simple functions, which completes the proof.

∗ Lemma 5.3 guarantees that if A ∈ B([0,T ]) and g ∈ G([0,T ]; U ) then 1A(·)g(·) ∈ G([0,T ]; U ∗). Lemma 5.4. If f : [0,T ] → L(U, V ) is weakly in G([0,T ]; U ∗) then for each A ∈ B([0,T ])

∗ 0 R ∗ 1 ∗ ∗ ZA : V → LP (Ω; ),ZAv := J A(·)f (·)v deﬁnes an inﬁnitely divisible cylindrical random variable with characteristic function Z ∗ C ∗ ∗ ∗ ϕZA : V → , ϕZA (v ) = exp Ψ(f (s)v ) ds . A

Furthermore, the cylindrical characteristics (bA, rA, νA) of ZA is given by Z ∗ ∗ ∗ ∗ bA : V → R, bA(v ) := a(f (s)v ) ds, (5.3) A Z ∗ ∗ ∗ ∗ ∗ ∗ rA : V → R, rA(v ) = hf (s)v , Qf (s)v i ds, (5.4) A −1 νA : Z(V ) → [0, ∞], νA = (µ ⊗ leb) ◦ χA , (5.5) where χA : [0,T ] × U → V is deﬁned by χA(s, u) := 1A(s)f(s)u.

∗ ∗ ∗ ∗ Proof. Since kf (·)v k∞ < ∞ for all v ∈ V , the uniform boundedness principle implies

∗ m := sup kf (s)kV ∗→U ∗ < ∞. (5.6) s∈[0,T ]

1 ∗ ∗ ∗ ∗ Consequently, we obtain k A(·)f (·)v k∞ 6 m kv k which implies the continuity of ZA : V → 0 R ∗ 0 R LP (Ω; ) by the continuity of J : G([0,T ]; U ) → LP (Ω; ). The form of the characteristic function follows immediately from (5.2).

The cylindrical random variable ZA is called the cylindrical integral of f on A. However, we want to deﬁne a genuine V -valued random variable as the stochastic integral of f which we achieve in the following way:

17 Deﬁnition 5.5. A function f : [0,T ] → L(U, V ) is called stochastically integrable w.r.t. L if ∗ 0 f is weakly in G([0,T ]; U ) and if for each A ∈ B([0,T ]) there exists an IA ∈ LP (Ω; V ) such that

∗ ∗ ∗ ∗ IA, v = ZAv for all v ∈ V , (5.7) where ZA denotes the cylindrical integral of f on A. In this case IA is denoted by Z f(s) dL(s) := IA. A

0 The existence of a random variable IA ∈ LP (Ω; V ) satisfying (5.7) is called that ZA is induced by IA. Such a random variable IA exists if and only if the cylindrical distribution of ZA extends to a measure, see [35, Th.IV.2.5]. Our approach by the cylindrical integral enables us to give sufficient and necessary conditions for the extension of the cylindrical distribution of ZA to a measure. In an arbitrary Banach space, as in the next Theorem, these conditions are rather abstract. However, in more specific spaces, such as Hilbert spaces or Banach spaces of type p ∈ [1, 2], some or even all of these conditions can be simplified significantly. We demonstrate this by a few subsequent corollaries.

Theorem 5.6. Let f : [0,T ] → L(U, V ) be a function which is weakly in G([0,T ]; U ∗). Then f is stochastically integrable if and only if the following is satisﬁed:

∗ (1) the mapping Ta is weak -weakly sequentially continuous where

∗ 1 ∗ ∗ ∗ Ta : V → L ([0,T ]; R),Tav = a(f (·)v ); (5.8)

(2) there exists a Gaussian covariance operator R: V ∗ → V satisfying

Z T hv∗, Rv∗i = hf ∗(s)v∗, Qf ∗(s)v∗i ds for all v∗ ∈ V ∗; (5.9) 0

(3) for χ: [0,T ] × U → V deﬁned by χ(s, u) := f(s)u, the cylindrical measure

ν : Z(V ) → [0, ∞], ν = (µ ⊗ leb) ◦ χ−1, (5.10)

extends to a measure and L´evymeasure on B(V ).

Note that although stochastic integrability of f requires that the cylindrical integral ZA is induced by a random variable IA for all A ∈ B([0,T ]), Conditions (5.9) and (5.10) can be considered as conditions only for A = [0,T ]. Condition (5.8) is the best sufficient and necessary prescription for the first term of the characteristics available, in order to guarantee integrability for the following reason: if L is a genuine Lévyprocess with classical characteristics (b, 0, 0) for some b ∈ U, then stochastic integrability of the function f according to Definition 5.5 reduces to Pettis integrability of the function f(·)b. In this case, a(·) = hb, ·i and Condition (5.8) is known to be equivalent to Pettis integrability of f(·)b, see [24, Th.4.1]. In Lemma 5.8 we explain this equivalence more general for genuine Lévy processes with arbitrary characteristics (b, Q, µ). In light of Lemma 5.4, Conditions (5.9) and (5.10) seem to be a straightforward conclu- sion, but the change from a cylindrical to a genuine infinitely divisible measure requires some further arguments. The correction term between the classical Lévy-Khintchine formula in (2.1) and the cylindrical version in (3.1) is considered in the following lemma.

18 Lemma 5.7. If ξ is a L´evymeasure on B(V ) then the function Z ∗ R ∗ ∗ 1 1 ∗ ∆ξ : V → , ∆ξ(v ) := hv, v i BV (v) − BR (hv, v i) ξ(dv), V

∗ ∗ ∗ ∗ is well deﬁned and satisﬁes ∆ξ(vn) → 0 for a sequence (vn)n∈N ⊆ V converging weakly to 0.

∗ ∗ ∗ ∗ Proof. Let v ∈ V and deﬁne D(v ) := {v ∈ V : |hv, v i| 6 1}. It follows for each v ∈ V that ∗ 1 1 ∗ ∗ 1 1 |hv, v i| | (v) − R (hv, v i)| = |hv, v i| ∗ c (v) + c ∗ (v) BV B BV ∩(D(v )) BV ∩D(v ) ∗ 2 |hv, v i| 1 (v) + 1 c (v). (5.11) 6 BV BV Proposition 5.4.5 in [17] gurantees Z Z ∗ 1 1 ∗ ∗ 2 c |hv, v i| | BV (v) − BR (hv, v i)| ξ(dv) 6 hv, v i ξ(dv) + ξ(BV ) < ∞, (5.12) V BV

∗ −1 ˜ For α := supn∈N kvnk define the mapping mα : V → V by mα(v) := α v. Then ξα := − −1 (ξ + ξ ) ◦ mα is a Lévymeasure on B(V ), and thus there exists an infinitely divisible ˜ 1 2 measure ϑ on B(V ) with characteristics (0, 0, ξα). The inequality 1 − cos(β) > 3 β for all ˜ |β| 6 1 implies by using the symmetry of ξα that the characteristic function of ϑ satisfies for each n ∈ N: Z ∗ ∗ − −1 ϕϑ(vn) = exp − 1 − cos(hv, vni) (ξ + ξ ) ◦ mα (dv) V Z −1 ∗ − 6 exp − 1 − cos(α hv, vni) (ξ + ξ )(dv) BV Z 2 ∗ 2 6 exp − 3α2 |hv, vni| ξ(dv) . BV ∗ Since the characteristic function ϕϑ is weakly sequentially continuous we obtain Z ∗ 2 3α2 ∗ |hv, vni| ξ(dv) 6 − 2 ln (ϕϑ(vn)) → 0 as n → ∞. BV

c ∗ c Since ξ(B ) < ∞ and 1 ∗ (v) |hv, v i| 1 for all n ∈ N and v ∈ B , Lebesgue’s theorem V D(vn) n 6 V of dominated convergence implies Z ∗ 1 ∗ (v) |hv, v i| ξ(dv) → 0 as n → ∞, D(vn) n c BV which completes the proof by (5.13).

19 Lemma 5.8. Let f : [0,T ] → L(U, V ) be a function which is weakly in G([0,T ]; U ∗). Then Condition (5.8) is satisfied if and only if ∗ ∗ ∗ (1) for every sequence (vn)n∈N ⊆ V converging weakly to 0 and A ∈ B([0,T ]) we have Z ∗ ∗ lim a(f (s)vn) ds = 0. (5.14) n→∞ A If L is a genuine Lévyprocess with characteristics (b, Q, µ) then its cylindrical characteristics (a, Q, µ) satisfies a = hb, ·i − ∆µ and (1) is equivalent to (2) the mapping t 7→ f(t)b is Pettis integrable. (5.15) Proof. Since a: U ∗ → R maps bounded sets to bounded sets according to (3.11), Condition (5.14) implies (5.8) by standard arguments. For the second part assume that L is a genuine Lévyprocess and we adopt the notation ∆µ from Lemma 5.7 but for the Lévymeasure µ on B(U). The first entries of the classical characteristics (b, Q, µ) and of the cylindrical characteristics (a, Q, µ) obey hb, u∗i = a(u∗) + ∗ ∗ ∗ ∗ ∗ ∞ ∆µ(u ) for all u ∈ U , which implies for all v ∈ V and h ∈ L ([0,T ]; R) the identity Z T Z T Z T ∗ ∗ ∗ ∗ ∗ h(s)hf(s)b, v i ds = h(s) a f (s)v ds + h(s) ∆µ f (s)v ds. (5.16) 0 0 0 ∗ ∗ ∗ Let m := sups∈[0,T ] kf (s)kV ∗→U ∗ be defined as in (5.6) and let (vn)n∈N ⊆ V be a sequence ∗ ∗ weakly converging to 0 with setting α := supn∈N kvnk. From (5.12) and [17, Pro.5.4.5] we conclude Z ∗ ∗ ∗ 2 c sup sup |∆µ(f (s)vn)| 6 sup hu, u i µ(du) + µ(BU ) < ∞. N ∗ n∈ s∈[0,T ] ku k6αm BU By applying Lebesgue’s theorem of dominated convergence and Lemma 5.7, we obtain for every h ∈ L∞([0,T ]; R) that

Z T Z T ∗ ∗ ∗ ∗ h(s)∆µ(f (s)vn) ds 6 khk∞ |∆µ(f (s)vn)| ds → 0. (5.17) 0 0

∗ 1 ∗ ∗ Let Sb : V → L ([0,T ]; R) denote the mapping defined by Sbv = hf(s)b, v i. It follows ∗ from (5.16) and (5.17) that Sb is weak -weakly sequentially continuous if and only if the ∗ mapping Ta, defined in (5.8), is weak -weakly sequentially continuous. Since the mapping ∗ t 7→ f(t)b is Pettis integrable if and only if Sb is weak -weakly continuous according to [24, Th.4.1] and since V is separable, the proof is completed. Proof. (Theorem 5.6). Sufficiency: according to (3.4) the cylindrical Lévyprocess L can be decomposed into L(t) = W (t) + P (t) for all t > 0, where W and P are independent, cylindrical Lévyprocesses with characteristics (0, Q, 0) and (a, 0, µ), respectively. For the cylindrical integral ZA of f on A ∈ B([0,T ]) we obtain ∗ W ∗ P ∗ ∗ ∗ ZAv = ZA v + ZA v for all v ∈ V ,

W P where ZA is the cylindrical integral w.r.t. W on A and ZA w.r.t. P on A. W Lemma 5.4 implies that the cylindrical random variable ZA is Gaussian with covariance rA deﬁned in (5.4). Since the covariance satisﬁes Z T ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ rA(v ) 6 hf (s)v , Qf (s)v i ds = hv , Rv i for all v ∈ V , 0

20 W Theorem 3.3 implies that the cylindrical distribution of ZA extends to a measure on B(V ). W 0 It follows from Theorem IV.2.5 in [35] that there exists a random variable IA ∈ LP (Ω; V ) W ∗ W ∗ ∗ ∗ with hIA , v i = ZA v for all v ∈ V . P According to Lemma 5.4 the cylindrical random variable ZA has the L´evymeasure νA, which obeys for every set C ∈ Z(V ) the inequality

Z Z Z T Z 1 1 νA(C) = C (f(s)u) ν(du) ds 6 C (f(s)u) ν(du) ds = ν(C). A U 0 U

Theorem 3.4 implies that νA extends to a L´evymeasure on B(V ), which is also denoted by νA. Thus, there exists a probability measure ϑA on B(V ) with characteristic function

Z ∗ ∗ ihv,v i ∗ 1 ∗ ∗ ϕϑA (v ) = exp e − 1 − ihv, v i BV (v) νA(dv) for all v ∈ V . (5.18) V Deﬁne the function Z ∗ R ∗ ∗ ∗ ∗ cA : V → , cA(v ) := a(f (s)v ) ds + ∆νA (v ), A

where the function ∆νA is deﬁned in Lemma 5.7. Let XA be a V -valued random variable with probability distribution ϑA. It follows from Lemma 5.4 and (5.18) that

P ∗ d ∗ ∗ ∗ ∗ ZA v = cA(v ) + hXA, v i for all v ∈ V , (5.19)

d P where = denotes equality in distribution. Linearity and continuity of ZA and hXA, ·i implies ∗∗ ∗ that cA ∈ V . It follows from Condition (5.8), Lemma 5.7 and Lemma 5.8 that cA is weakly ∗ sequentially continuous. Since V is separable, we obtain that cA is weakly continuous ([19,

Co.2.7.10]), which implies cA ∈ V . Consequently, we can define the Dirac measure δcA at P the point cA ∈ V . It follows from (5.19) that the cylindrical distribution of ZA extends to the convolution δcA ∗ ϑA. Consequently, [35, Th.IV.2.5] guarantees that there exists a P 0 P ∗ P ∗ ∗ ∗ random variable IA ∈ LP (Ω; V ) with hIA , v i = ZA v for all v ∈ V , which shows the stochastic integrability of f. 0 Necessity: let IA ∈ LP (Ω; V ) denote the stochastic integral of f on A and let (cA,SA, ξA) be the characteristics of the infinitely divisible random variable IA. Then, due to the unique- ness of the Lévy-Khintchine formula in R, for each v∗ ∈ V ∗ the characteristics of the real ∗ ∗ valued random variables hIA, v i and ZAv coincide which results in Z ∗ ∗ ∗ ∗ hcA, v i − ∆ξA (v ) = a (f (s)v ) ds, (5.20) A Z ∗ ∗ ∗ ∗ ∗ ∗ hv ,SAv i = hf (s)v , Qf (s)v i ds, (5.21) A ∗ −1 −1 ∗ −1 ξA ◦ (v ) = µ ⊗ leb ◦ χA ◦ (v ) . (5.22)

∗ Here, we obtain the characteristics of hIA, v i on the left hand side by a standard calculation for the transform of an inﬁnitely divisible measure under a linear mapping (see e.g. [33, ∗ Pro.11.10]), whereas the characteristics of ZAv on the right hand side is given in Lemma 5.4. Equation (5.20) shows Condition (5.8) due to Lemma 5.7 and Lemma 5.8. By choosing A := [0,T ], identity (5.21) implies Condition (5.9).

21 By using (5.22) and [33, Pro.11.10], it follows from linearity that the Rn-valued inﬁnitely ∗ ∗ ∗ ∗ divisible random variables hIA, v1 i,..., hIA, vni and ZAv1 ,...,ZAvn) have the same L´evy ∗ ∗ ∗ N measures for all v1 , . . . , vn ∈ V and n ∈ , i.e.

−1 −1 −1 ξA ◦ π ∗ ∗ = µ ⊗ leb ◦ χ ◦ π ∗ ∗ . v1 ,...,vn A v1 ,...,vn

−1 Consequently, by choosing A = [0,T ] we have ν = (µ ⊗ leb) ◦ χA and the image cylindrical measure ν extends to the Lévymeasure ξA. Remark 5.9. In the work [32] together with van Gaans, we developed a stochastic integral for deterministic integrands w.r.t. martingale valued measures in Banach spaces, i.e. in particular with respect to the compensated Poisson random measure of a classical Lévy process in a Banach space U. The integrability of a function is described in terms of the appropriate convergence of a random series, which is very similar to the case of γ-radonifying operators. This approach cannot be applied to cylindrical Lévyprocesses as they do not satisfy an Itô-Lévydecomposition in the underlying Banach space but only in R, which is then depending on the argument in U ∗. Both integrals in the current work and in [32] underly a kind of a stochastic version of the Pettis integral. As mentioned before, we simplify the conditions in Theorem 5.6 in some more specific spaces. We begin with the most important case of a Hilbert space.

Theorem 5.10. Assume that V is a Hilbert space with orthonormal basis (ek)k∈N and let f : [0,T ] → L(U, V ) be a function which is weakly in G([0,T ]; U ∗). Then f is stochastically integrable if and only if the following is satisﬁed:

(1) the mapping Ta is weak-weakly sequentially continuous where

∗ 1 ∗ ∗ ∗ Ta : V → L ([0,T ]; R),Tav = a(f (·)v ); (5.23)

Z T (2) tr [f(s)Qf ∗(s)] ds < ∞; (5.24) 0 Z T Z n ! X ∗ 2 (3) lim sup sup hu, f (s)eki ∧ 1 µ(du) ds = 0. (5.25) m→∞ n m > 0 U k=m Proof. The closed graph theorem shows that

Z T hv∗, Rw∗i = hf ∗(s)v∗, Qf ∗(s)w∗i ds for all v∗, w∗ ∈ V ∗, 0 defines a positive, symmetric and bounded operator R: V ∗ → V . By applying Tonelli’s theorem we can conclude that the operator R is of trace class if and only if Condition (5.24) is satisfied. Since the space of Gaussian covariance operators in Hilbert spaces coincide with the space of trace class operators by [35, Th.IV.2.4], we have established the equivalence of Conditions (5.9) and (5.24). If f is stochastically integrable then Theorem 5.6 implies that the cylindrical measure ν, defined in (5.10), extends to a measure and it is a Lévymeasure in V . Since V is a Hilbert space, the latter implies (see [25, Th.VI.4.10]), that Z kvk2 ∧ 1 ν(dv) < ∞, V

22 which shows Condition (5.25). It remains to show that (5.25) implies Condition (5.10), for which we can assume that the cylindrical characteristics of L is of the form (a, 0, µ). We deﬁne the space Vn := span{e1, . . . , en} and we denote by πn : V → V the orthogonal projection on Vn for each n ∈ N. Let ZA denote the cylindrical integral of f on A ∈ B([0,T ]), which has the characteristics 0 ˜ (bA, 0, νA) according to Lemma 5.4. If ZA denotes an independent copy of ZA then ZA := 0 − ZA + ZA is a cylindrical random variable with characteristics (0, 0, νA + νA ). Since πn is ˜ ∗ a Hilbert-Schmidt operator, the cylindrical distribution of ZA ◦ πn extends to a probability measure ϑn on B(V ) due to [35, Th.VI.5.2], which is inﬁnitely divisible with characteristics − −1 2 (0, 0, ξn) where ξn := (νA + νA ) ◦ πn . By using the inequality 1 − cos β 6 2(β ∧ 1) for all β ∈ R we obtain for every v∗ ∈ V that Z ∗ ∗ 1 − ϕϑn (v ) = 1 − exp cos(hv, v i) − 1 ξn(dv) V Z Z ∗ ∗ 2 6 1 − cos(hv, v i) ξn(dv) 6 2 hv, v i ∧ 1 ξn(dv). V V m Let gm denote the density of the standard normal distribution on B(R ). For every m, n ∈ N with m 6 n it follows that Z 1 − Re ϕϑn (βmem + ··· + βnen) gn−m+1(βm, . . . , βn) dβm ··· dβn Rn−m+1  n 2  Z Z X 6 2  βkhv, eki ∧ 1 ξn(dv) gn−m+1(βm, . . . , βn) dβm ··· dβn Rn−m+1 V k=m

  n 2   Z Z X 6 2   βkhv, eki  gn−m+1(βm, . . . , βn) dβm ··· dβn ∧ 1 ξn(dv) n−m+1 V R k=m Z n ! X 2 = 2 hv, eki ∧ 1 ξn(dv) V k=m Z n ! X 2 −1 = 2 hπnv, eki ∧ 1 (vA + vA )(dv) V k=m Z n ! X 2 = 4 hv, eki ∧ 1 v(dv). V k=m Condition (5.25) implies Z lim sup sup 1 − Re ϕϑn (βmem + ··· + βnen) gn−m+1(βm, . . . , βn) dβm ··· dβn = 0, n−m+1 m→∞ n>m R which shows by [25, Le.VI.2.3] that the family {ϑn}n∈N is relatively compact in M(V ). Since Z˜ and thus its characteristic function ϕ are continuous (see [35, Pro.IV.3.4]), we A Z˜A have for each v∗ ∈ V ∗:

∗ ∗ ∗ ∗ lim ϕϑ (v ) = lim ϕ ˜ he1, v ie1 + ··· + hen, v ien = ϕ ˜ (v ). (5.26) n→∞ n n→∞ ZA ZA

Together with the relative compactness of {ϑn}n∈N it follows from Theorem IV.3.1 in [35] that the probability measures {ϑn}n∈N converges weakly to a measure ϑ, which coincides

23 with the cylindrical distribution of ZÃ on Z(V ). Consequently, [35, Th.IV.2.5] guarantees ˜ 0 ˜ ∗ ˜ ∗ ∗ ∗ that there exists a random variable IA ∈ LP (Ω; V ) with hIA, v i = ZAv for all v ∈ V . − ˜ ˜ Thus, the cylindrical Lévymeasure νA + νA of ZA extends to the Lévymeasure of IA. Since

− νA(C) 6 νA(C) + νA (C) for all C ∈ Z(V ),

Theorem 3.4 implies that νA extends to a Lévymeasure which shows Condition (5.10). Remark 5.11. In the work [9] on Lévyprocesses in Hilbert spaces, the author developes among others a theory of stochastic integration for deterministic operators with respect to a classical Lévyprocess in a separable Hilbert space U. More specifically, let V be another separable Hilbert space and define the set

( Z T ) 2 S := f : [0,T ] → L(U, V ): f is strongly measurable and kf(s)kU→V ds < ∞ . 0

Then by using tighness conditions for infinitely divisible measures in Hilbert spaces (see [25]), a stochastic integral is defined for integrands in S in [9]. In this case of genuine Lévy processes, it is easy to see that each function f ∈ S satisfies Condition (5.15) in Lemma 5.8 and Conditions (5.24) and (5.25) in Theorem 5.10. Thus, Theorem 5.10 guarantees that each f ∈ S is stochastically integrable in our sense according to Definition 5.5. Corollary 5.12. Under the assumption of Theorem 5.6 let the cylindrical measure ν be defined by (5.10). Then we have the following: (a) If V is of type p ∈ [1, 2], then Condition (5.10) replaced by

0 R p (3 ) ν extends to a measure and V kvk ∧ 1 ν(dv) < ∞, implies together with (5.8) and (5.9) that f is stochastically integrable. (b) If V is of cotype q ∈ [2, ∞) then stochastic integrability of f implies

0 R q (3 ) ν extends to a measure and V kvk ∧ 1 ν(dv) < ∞. Proof. The reformulation of Condition (5.10) in Theorem 5.6 follows in both cases from results in the article [10]. In this work the authors establish sufficient conditions in Banach spaces of type p and necesssary conditions in Banach spaces of cotype q for a σ-finite measure to be a Lévymeasure. Corollary 5.13. If V is the space `p(R) of sequences for p ∈ [2, ∞) equipped with the standard basis (ek)k∈N, then Condition (5.9) and (5.10) in Theorem 5.6 can be replaced by

p/2 ∞ Z T ! 0 X ∗ ∗ (2 ) hf (s)ek, Qf (s)eki ds < ∞. (5.27) k=1 0 (30) the cylindrical measure ν extends to a Radon measure on B(V ) and satisﬁes Z (a) (kvkp ∧ 1) ν(dv) < ∞; (5.28) V ∞ Z !p/2 X 2 (b) |hv, eki| ν(dv) < ∞. (5.29) k=1 kvk61

24 Proof. Theorem 5.6 in [35] guarantees that Conditions (5.9) and (5.27) are equivalent. The class of L´evymeasures in `p(R) is described by Conditions (5.28) and (5.29) according to a result in [16].

6 Ornstein-Uhlenbeck processes

In this last part we apply the previous developed theory of stochastic integration to define Ornstein-Uhlenbeck processes driven by cylindrical Lévyprocesses. These processes are important since they are solutions of stochastic evolution equations driven by cylindrical Lévyprocesses, see for instance [26]. We do not study this connection in this work but we consider examples of specific cases, then these processes exist. If L is a cylindrical Lévyprocess in a Banach space U and G ∈ L(U, V ) then the ∗ ∗ ∗ ∗ ∗ cylindrical Lévyprocess GL defined by (GL)(t)v := L(t)(G v ) for all v ∈ V and t > 0 is a cylindrical Lévyprocess in V . It follows for a function f : [0,T ] → L(U, V ) that if f(·) ◦ G is stochastically integrable w.r.t. L then f is stochastically integrable w.r.t GL and Z T Z T f(s)G dL(s) = f(s) d(GL)(s). 0 0 Thus, without loss of generality we can assume in this section U = V , i.e. L is a cylindrical Lévyprocess in the separable Banach space V .

Deﬁnition 6.1. If a strongly continuous semigroup (T (t))t∈[0,T ] on V is stochastically integrable with respect to L, then we call the stochastic process (X(t): t ∈ [0,T ]) deﬁned by Z t X(t) := T (t)v0 + T (t − s) dL(s) for all t ∈ [0,T ], 0

Ornstein-Uhlenbeck process with initial value v0 ∈ V driven by L. The existence of the stochastic convolution integral in Deﬁnition 6.1 is guaranteed by the following result. Lemma 6.2. A function f : [0,T ] → L(V,V ) is stochastically integrable if and only if f(T −·) is stochastically integrable. In this case we have the equality in distribution: Z T Z T f(s) dL(s) =d f(T − s) dL(s). 0 0 Proof. If f is weakly in G([0,T ]; V ∗) then f(T − ·) is also weakly in G([0,T ]; V ∗). Since Conditions (5.8) - (5.10) in Theorem 5.6 are invariant under a transformation s 7→ T − s the ﬁrst part of the Lemma is proved. The identity

Z T ! Z T ! exp Ψ(f ∗(s)v∗) ds = exp Ψ(f ∗(T − s)v∗) ds for all v∗ ∈ V ∗, 0 0 shows the equality of the distributions by Lemma 5.4 As an example of an Ornstein-Uhlenbeck process we consider the case of a diagonalisable semigroup and of a cylindrical L´evyprocess deﬁned by a sum acting independently along the eigenbasis of the semigroup, cf. Lemma 4.2. This kind of setting is considered in several publications, e.g. [27] and [28].

25 Corollary 6.3. Assume that V is a Hilbert space and that there exists an orthonormal basis (ek)k∈N of V and (γk)k∈N ⊆ R such that the semigroup (T (t))t∈[0,T ] satisﬁes

∗ γkt T (t)ek = e ek for all t ∈ [0,T ], k ∈ N . (6.1) Let the cylindrical Lévyprocess L be of the form ∞ ∗ X ∗ ∗ ∗ L(t)v = hek, v i`k(t) for all v ∈ V , t > 0, k=1 where (`k)k∈N is a sequence of independent, symmetric Lévyprocesses in R with characteristics (0, 0, νk). Then the semigroup (T (t))t∈[0,T ] is stochastically integrable w.r.t. L if and only if ∞ Z T Z X 2λks 2 e |β| ∧ 1 νk(dβ) ds < ∞. (6.2) R k=1 0 Proof. According to Lemma 4.2, the cylindrical Lévyprocess L has characteristics (0, 0, µ) −1 N satisfying µ◦ek (dβ) = νk(dβ) for all k ∈ . Independence of the Lévyprocesses (`k)k∈N implies for the Lévymeasure µ ◦ π−1 of L(1)(e ),...,L(1)(e ) for 0 m n the em,...,en m n 6 6 identity n −1 X −1 µ ◦ πe ,...,e = δ0 ⊗ · · · ⊗ δ0 ⊗ µ ◦ πe ⊗ δ0 ⊗ · · · ⊗ δ0 . m n | {z } k | {z } k=m k−m times n−k times Thus, we obtain for all v ∈ V : Z T Z n ! X ∗ 2 hv, T (s)eki ∧ 1 µ(dv) ds 0 V k=m Z T Z n ! X λks 2 = hv, e eki ∧ 1 µ(dv) ds 0 V k=m T n ! Z Z X 2 = eλksβ ∧ 1 µ ◦ π−1 (dβ ··· dβ ) ds k em,...,en m n Rn−m+1 0 k=m n Z T Z X 2λks 2 −1 = e β ∧ 1 µ ◦ ek (dβ) ds. R k=m 0 ∗ Since V is a Hilbert space, the adjoint semigroup (T (t))t∈[0,T ] is strongly continuous, and ∗ thus (T (t))t∈[0,T ] is weakly in G([0,T ]; V ). An application of Theorem 5.10 establishes that the semigroup is stochastically integrable if and only if (6.2) is satisfied. Example 6.4. Assume that the cylindrical Lévyprocess L is given as in Example 4.5 by `k(·) := σkhk(·), where (hk)k∈N is a sequence of independent, symmetric, α-stable processes and (σk)k∈N ⊆ R. If the strongly continuous semigroup (T (t))t∈[0,T ] satisfies (6.1) for λk < 0 with λk → −∞ for k → ∞, then a simple calculation shows that (6.2) is satisfied if and only if

∞ α X |σk| < ∞. |γk| k=1 In this case, the semigroup is stochastically integrable, which coincide with a result in [27].

26 We now consider stochastic integrability of a semigroup (T (t))t>0 w.r.t. to a cylindrical L´evynoise constructed by subordination as in Example 4.7. In fact, we will show integrability in a possible smaller subspace E ⊆ V with norm k·kE assuming T (t)(V ) ⊆ E for almost all t > 0. Recall that HC denotes the reproducing kernel Hilbert space of the subordinated cylindrical Wiener process and iC : HC → V its embedding. In the following denote by R(HC ,E) the space of γ-radonifying operators g : HC → E. For g ∈ R(HC ,E) and p ∈ [1, ∞) deﬁne

" ∞ p # p X kgk := E γkghk , Rp(HC ,E) k=1 E where (γk)k∈N is a family of independent, real valued standard normally distributed random variables and (hk)k∈N is an orthonormal basis in HC . Corollary 6.5. Let L be the cylindrical L´evyprocess in a separable Banach space V deﬁned by

∗ ∗ ∗ ∗ L(t)v := W `(t) v for all v ∈ V , t > 0, (6.3) where W denotes a cylindrical Wiener process with covariance operator C and ` is a real valued subordinator with characteristics (0, 0, ρ). Let E be a separable Banach space of type ∗ p ∈ [1, 2]. If the semigroup (T (t))t∈[0,T ] in V is weakly in G([0,T ]; V ) and the mapping T (t) ◦ iC is in R(HC ,E) for almost all t ∈ [0,T ], then

Z ∞ Z T |r|p/2 kT (s) ◦ i kp ∧ 1 ds ρ(dr) < ∞, C Rp(HC ,E) 0 0 implies that the semigroup (T (t))t∈[0,T ] is stochastically integrable w.r.t. L and the stochastic integral is E-valued.

Proof. Let γ denote the canonical√ cylindrical Gaussian measure on HC and let κ be deﬁned as in Lemma 4.8, i.e. κ(h, s) := s iC h, and χ as in Theorem 5.6, i.e. χ(s, v) := T (s)v. For applying Corollary 5.12 we have to show that the cylindrical measure ν = (γ ⊗ ρ) ◦ κ−1 ⊗ leb ◦ χ−1 extends to a measure on B(V ). For this purpose deﬁne the family of cylindrical sets

∗ ∗ ∗ ∗ G : = C(v1 , . . . , vn; R): v1 , . . . , vn ∈ V , N R = (a1, b1) × · · · × (an, bn), −∞ 6 aj < bj 6 ∞, j = 1, . . . , n, n ∈ ,

∗ ∗ ∗ where V denotes the weak dense subspace of V such that (T (t))t∈[0,T ] acts strongly continuously on V . Since V is separable and V separates points in V , Theorem I.2.1 in −1 [35] guarantees that G generates the σ-algebra B(V ). Deﬁne γs := γ ◦ T (s) ◦ iC for all 0 R s ∈ [0,T ]. Let Γ: HC → LP (Ω; ) be a cylindrical random variable with distribution γ. ∗ ∗ ∗ ∗ ∗ Since Γ(T (sk)v ) → Γ(T (s)v ) in probability for sk → s and all v ∈ V , the portmanteau Rn ∗ ∗ theorem in implies for each C := C(v1 , . . . , vn; R) ∈ G:

∗ ∗ ∗ ∗ γs (C) = P (Γ(T (sk)v ),..., Γ(T (sk)v )) ∈ R k 1 n (6.4) ∗ ∗ ∗ ∗ → P (Γ(T (s)v1 ),..., Γ(T (s)vn)) ∈ R = γs(C) as sk → s.

Consequently, Theorem 452C in [13] on disintegration of measures implies that s 7→ γs(B) is measurable for all B ∈ B(V ) and almost all s ∈ [0,T ], and that there exists a Borel measure

27 ϑ on B(V ) such that

Z T ϑ(B) = γs(B) ds for all B ∈ B(V ). (6.5) 0

−1/2 Deﬁne for each r ∈ (0, ∞) the measure ϑr(B) := ϑ(r B) for all B ∈ B(V ). If a sequence (rk)k∈N ⊆ (0, ∞) converges to r ∈ (0, ∞) then it follows similarly as in (6.4) that −1/2 −1/2 γs(rk C) → γs(r C) for all s ∈ [0,T ] and C ∈ G. By applying Lebesgue’s theorem of dominated convergence we conclude from (6.5) that the mapping r 7→ ϑr(C) is continuous for all C ∈ G. Another application of Theorem 452C in [13] implies that there exists a measure µ on B(V ) satisfying Z ∞ Z ∞ −1/2 µ(B) = ϑr(B) ρ(dr) = ϑ(r B) ρ(dr) for all B ∈ B(V ). 0 0

−1/2 Note that by our argument above the function (s, r) 7→ γs(r C) is separately continuous in both variables for all C ∈ G and thus jointly measurable. Tonelli’s theorem enables us to conclude for all C ∈ G that Z ∞ Z T Z T Z ∞ −1/2 −1/2 µ(C) = γs(r C) ds ρ(dr) = γs(r C) ρ(dr) ds = ν(C), 0 0 0 0 which shows that ν extends to the measure µ on B(V ). The measure µ satisﬁes

Z Z T Z ∞ Z p √ p kvk ∧ 1 µ(dv) = T (s)( r iC h) ∧ 1 γ(dh)ρ(dr)ds V 0 0 HC Z T Z ∞ Z p/2 p 6 |r| kT (s)(iC h)k γ(dh) ∧ 1 ρ(dr)ds 0 0 HC Z T Z ∞ = |r|p/2 kT (s) ◦ i kp ∧ 1 ρ(dr) ds. C Rp(HC ,E) 0 0 An application of Corollary 5.12 completes the proof.

A very similar result as Corollary 6.5 is derived in [8]. However, the conditions in our result are purely intrinsic, whereas the result in [8] is based on conditions in terms of an additional Banach space, which is not related to the problem under consideration.

Acknowledgments: the author would like to thank the referees for their careful reading and valuable comments and suggestions.

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